TY - JOUR
T1 - Evolutionary games on the lattice
T2 - death and birth of the fittest
AU - Foxall, Eric
AU - Lanchier, Nicolas
N1 - Funding Information:
Acknowledgment. Eric Foxall was supported in part by an NSERC PDF Award and Nicolas Lanchier by NSA Grant MPS-14-040958.
Publisher Copyright:
© 2017. Alea (Rio de Janeiro). All rights reserved.
PY - 2017
Y1 - 2017
N2 - This paper investigates the long-term behavior of an interacting particle system of interest in the hot topic of evolutionary game theory. Each site of the d- dimensional integer lattice is occupied by a player who is characterized by one of two possible strategies. Following the traditional modeling approach of spatial games, the configuration is turned into a payoff landscape that assigns a payoff to each player based on her strategy and the strategy of her 2d neighbors. The payoff is then interpreted as a fitness, by having each player independently update their strategy at rate one by mimicking their neighbor with the largest payoff. With these rules, the mean-field approximation of the spatial game exhibits the same asymptotic behavior as the popular replicator equation. Except for a coexistence result that shows an agreement between the process and the mean-field model, our analysis reveals that the two models strongly disagree in many aspects, showing in particular that the presence of a spatial structure in the form of local interactions plays a key role. More precisely, in the parameter region where both strategies are evolutionary stable in the replicator equation, in the spatial model either one strategy wins or the system fixates to a configuration where both strategies are present. In addition, while defection is evolutionary stable for the prisoner’s dilemma game in the replicator equation, space favors cooperation in our model.
AB - This paper investigates the long-term behavior of an interacting particle system of interest in the hot topic of evolutionary game theory. Each site of the d- dimensional integer lattice is occupied by a player who is characterized by one of two possible strategies. Following the traditional modeling approach of spatial games, the configuration is turned into a payoff landscape that assigns a payoff to each player based on her strategy and the strategy of her 2d neighbors. The payoff is then interpreted as a fitness, by having each player independently update their strategy at rate one by mimicking their neighbor with the largest payoff. With these rules, the mean-field approximation of the spatial game exhibits the same asymptotic behavior as the popular replicator equation. Except for a coexistence result that shows an agreement between the process and the mean-field model, our analysis reveals that the two models strongly disagree in many aspects, showing in particular that the presence of a spatial structure in the form of local interactions plays a key role. More precisely, in the parameter region where both strategies are evolutionary stable in the replicator equation, in the spatial model either one strategy wins or the system fixates to a configuration where both strategies are present. In addition, while defection is evolutionary stable for the prisoner’s dilemma game in the replicator equation, space favors cooperation in our model.
KW - Interacting particle systems
KW - cooperation
KW - evolutionary game theory
KW - evolutionary stable strategy
KW - prisoner’s dilemma
KW - replicator equation
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U2 - 10.30757/ALEA.v14-16
DO - 10.30757/ALEA.v14-16
M3 - Article
AN - SCOPUS:85069990741
SN - 1980-0436
VL - 14
SP - 271
EP - 298
JO - Alea
JF - Alea
IS - 1
ER -